Showing papers on "Undecidable problem published in 1982"
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TL;DR: The syntactic structure of sets of ancestors and sets of descendants is considered, as well as that of unions of congruence classes, taken over (infinite) context-free languages or regular sets.
113 citations
Book•
01 Jan 1982
TL;DR: This book discusses Computable Properties of Sets, Alternative Approaches to Computability, and Godel's Incompleteness Theorem.
Abstract: 1 Introduction.- 1.1 Partial Functions and Algorithms.- 1.2 An Invitation to Computability Theory.- 1.3 Diagonalization and the Halting Problem.- 2 The Syntax and Semantics of while-Programs.- 2.1 The Language of while-Programs.- 2.2 Macro Statements.- 2.3 The Computable Functions.- 3 Enumeration and Universality of the Computable Functions.- 3.1 The Effective Enumeration of while-Programs.- 3.2 Universal Functions and Interpreters.- 3.3 String-Processing Functions.- 3.4 Pairing Functions.- 4 Techniques of Elementary Computability Theory.- 4.1 Algorithmic Specifications.- 4.2 The s-m-n Theorem.- 4.3 Undecidable Problems.- 5 Program Methodology.- 5.1 An Invitation to Denotational Semantics.- 5.2 Recursive Programs 110 5.3* Proof Rules for Program Properties.- 6 The Recursion Theorem and Properties of Enumerations.- 6.1 The Recursion Theorem.- 6.2 Model-Independent Properties of Enumerations.- 7 Computable Properties of Sets (Part 1).- 7.1 Recursive and Recursively Enumerable Sets.- 7.2 Indexing the Recursively Enumerable Sets.- 7.3 Godel's Incompleteness Theorem.- 8 Computable Properties of Sets (Part 2).- 8.1 Rice's Theorem and Related Results.- 8.2 A Classification of Sets.- 9 Alternative Approaches to Computability.- 9.1 The Turing Characterization.- 9.2 The Kleene Characterization.- 9.3 Symbol-Manipulation Systems and Formal Languages.- References.- Notation Index.- Author Index.
76 citations
TL;DR: The decidability of the emptmess problem for Q is useful in proving the solvabdity of some number-theoreuc problems and can be used to prove that the language L cannot be accepted by any machme in Q.
Abstract: Let Q be the class of deterministic two-way one-counter machines accepting only bounded languages. Each machine in Q has the property that in every accepting computation, the counter makes at most a fixed number of reversals. We show that the emptiness problem for Q is decidable. When the counter is unrestricted or when the machine is provided with two reversal-bounded counters, the emptiness problem becomes undecidable. The decidability of the emptiness problem for Q is useful in proving the solvability of some numbertheoretic problems. It can also be used to prove that the language L = {u1iu2i2|i≥0} cannot be accepted by any machine in Q (u1 and u2 are distinct symbols). The proof technique is new in that it does not employ the usual "pumping", "counting", or "diagonal" argument. Note that L can be accepted by a deterministic two-way machine with two counters, each of which makes exactly one reversal.
42 citations
TL;DR: In this paper, the authors obtained similar theorems of this type but the winning conditions are extremely simple relations (polynomial equations) and the winning strategies are computable.
Abstract: Computing machines using algorithms play games and even learn to play games. However, the inherent finiteness properties of algorithms impose limitations on the game playing abilities of machines. M. Rabin illustrated this limitation in 1957 by constructing a two-person win-lose game with decidable rules but no computable winning strategies. Rabin's game was of the type where two players take turns choosing integers to satisfy some decidable but very complicated winning condition. In the present paper we obtain similar theorems of this type but the winning conditions are extremely simple relations (polynomial equations). Specific examples are given.
24 citations
TL;DR: The Paris-Harrington Theorem is a variant of the Finite Ramsey Theorem, which, though provable in ordinary set theory (and so a theorem of ordinary mathematical practice), is not provable for formal number theory as mentioned in this paper.
Abstract: Publisher Summary This chapter discusses the varieties of arboreal experience. The Paris–Harrington Theorem is a variant of the Finite Ramsey Theorem, which, though provable in ordinary set theory (and so a theorem of ordinary mathematical practice), is not provable in formal number theory. This independence being of a character different from that of Godel's undecidable sentence or that associated with independence results in set theory, the result has been much touted by logicians seeking social acceptance by the rest of the mathematical world as something closely akin to the coming of the millennium. This is unfortunate as the result really is remarkable philosophically because of this new character of independence and mathematically for reasons requiring a new paragraph. The difference between the usual Finite Ramsey Theorem and the Paris–Harrington Theorem is surprisingly consequential, concerning computationally hairy combinatorial problems with unfeasible bounds; but the former's unfeasible bounds are readily intelligible, while the latter's are not. The relation between Kruskal's Theorem and ordinals is fairly close. In fact, Kruskal's Theorem is most memorably stated, if not in terms of ordinals and well-ordering, at least in terms of well-quasi-ordering.
24 citations
Book•
01 Jan 1982
TL;DR: The theory of undecidable sentences as discussed by the authors was presented as clearly and as rigorously as possible in the first part of the book. But it was only used in the final chapter and the appendix.
Abstract: The famous theory of undecidable sentences created by Kurt Godel in 1931 is presented as clearly and as rigorously as possible. Introductory explanations beginning with the necessary facts of arithmetic of integers and progressing to the theory of representability of arithmetical functions and relations in the system (S) prepare the reader for the systematic exposition of the theory of Godel which is taken up in the final chapter and the appendix.
16 citations
TL;DR: In this paper, the authors give brief proofs of two results on undecidability of a first-order theory using a construction which they call a modified Boolean power, which was introduced by Burris in late 1978, and succeeded in using this construction to prove the results in this paper.
Abstract: In this paper we will give brief proofs of two results on the undecidability of a first-order theory using a construction which we call a modified Boolean power. Modified Boolean powers were introduced by Burris in late 1978, and the first results were announced in [2]. Subsequently we succeeded in using this construction to prove the results in this paper, namely Ershov's theorem that every variety of groups containing a finite non-abelian group has an undecidable theory, and Zamjatin's theorem that a variety of rings with unity which is not generated by finitely many finite fields has an undecidable theory. Later McKenzie further modified the construction mentioned above, and combined it with a variant of one of Zamjatin's constructions to prove the sweeping main result of [3]. The proofs given here have the advantage (over the original proofs) that they use a single construction.
3 citations
TL;DR: The complexity of the equivalence problem for several classes of simple programs with a fixed number of program variables is investigated and the classes are shown to have undecidable, NP-hard, or polynomial-time decidable equivalence problems.
2 citations
Proceedings Article•
01 Jun 1982
TL;DR: The authors shall use the model of Communicating Sequential Processes (CSP) developed by Hoare to investigate the detection of inherent deadlocks in distributed programs.
Abstract: : In this paper, the concept of 'inherent deadlock' in distributed programs is defined Several algorithms for detecting inherent deadlocks are given Deadlock prevention is crucial in distributed programs In order to ensure the correctness of a distributed program, we must avoid the occurrence of the deadlock in its execution Unfortunately, the deadlock problem in distributed program is undecidable, as the halting problem in the sequential problem However, partial solutions to the deadlock problem exist In this the authors shall investigate the detection of inherent deadlocks in distributed programs There are many models for distributed programs In this paper, the authors shall use the model of Communicating Sequential Processes (CSP) developed by Hoare In section 2 they develop some simplifications and abstractions of CSP and define the concept of 'inherent deadlock' They solve its decision problem In section 3 the authors define the concept of D-execution, and obtain a sufficient condition and a corresponding algorithm for detecting inherent deadlock In sections 4 and 5 the authors introduce the concept 'matching number' as the foundation for obtaining two sufficient conditions for detecting inherent deadlock Then they reduce these conditions to the solvability of some kind of indeterminate equation and give its decision algorithm
2 citations
TL;DR: In SIGP LAN Notices I 7:2 K.-C.
Abstract: In SIGP LAN Notices I 7:2 K.-C. Tai mentioned that no algorithm is known which could determine whether the result of a given program depends on the mechanism of parameter passing, being different when parameters are passed by reference or by copy. Indeed, the problem is undecidableo Let us assume the contrary, that there would algorithm. By Church~s thesis; we could write be such an function DEPENDS(FUNCT, DATA: in STRING) return BOOLEAN such that when FUNCT is a piece of s,)urce program text tha ~ defines a function with one STRING-typed parameter, DEPENDS returns TRUE iff the result of FUNCT applied to DATA depends On the mechanism of parameter passing. The function DEPENDS should itself be implementation-independent.
2 citations